We recently finished up a section on area and volume using integrals and we were learning about what the anti-derivative is. My question is I am not sure if there is proof that the anti-derivative can give us the area or volume. I asked my teacher about this but there doesn't seem to be a concrete proof? I am just wondering how did Newton or what not come to learn that the anti-derivative, which I see as the pre-slope leads to the area and volume?
Right now I am a skeptic about the anti derivative. Can someone make me a believer? I need some proof. I am just not convinced.